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| Nonunital positive Linear Maps on C*-Algebras |
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Citation： |
Li Bingren.Nonunital positive Linear Maps on C*-Algebras[J].Chinese Annals of Mathematics B,1985,6(1):1~4 |
| Page view： 808
Net amount： 843 |
Authors： |
Li Bingren; |
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| Abstract： |
In many cases, studying positive linear maps on C^*-algebras, we always assume that it is unital(i. e. it carries identity to identity). In this paper, the author discusses nonunital positive linear maps on C^*-algebras. First, similar to positive functionals on A, if Ф is a positive linear map from C^*-algebra A into B, then
$[{\rm{||}}\Phi {\rm{|| = sup\{ ||}}\Phi {\rm{(}}a){\rm{|| |}}a \in {A_ + },||a|| \le 1\} = \mathop {\lim }\limits_l ||\Phi ({v_l})|| = \mathop {\lim }\limits_l ||\Phi (v_l^2)||\]$
where {v_i} is an approximate identity of A. Then the author proves that if Ф is an n-positive linear map from A into B, and ||Ф|| =1, then Ф can be extended to an unital n-positive linear map [\bar Ф] from A+I_A into B+I_B(or into B, if B has identity). This result can also be used to generalize some results about unital positive maps. |
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